Morning from Lijiang!
I just upgraded my OS from 20.04 to 22.04 yesterday morning, and I bumped into the same problem. (The upgrade itself was due to my failed attempt to use NVDEC-based hardware-decoding for Firefox/FFMPEG, but this was another story.)
I tried the anthy solution from the answer. But ibus-anthy is not very intelligient compared to ibus-mozc. For example, it is not very convenient to input 頭高型 (あたまだかがた) or 菅義偉 (すがよしひで), and the companion kasumi dictionary editor for anthy never works for me on my system.
Lucikly, I finally made it and upgrade mozc to a version that works with Anki on my fresh 22.04 last night. Here was how I did it, but remember, YMMV!
I first thought it was an Anki issue (since only Anki observed this problem and other apps just worked fined) and upgrade Anki to the latest version (2.1.65 Qt6) from the official website. But to no avail, this does not solve my mozc problem.
However, I also found out from this issue comment that the problem was already solved by a newer version of mozc. Then I tried to upgrade mozc to a newer version. I used the deb approach, since I don't want to compile everything from source. (I am an ex-programmer now.) I used this Deiban webpage to download the necessary deb files. I downloaded ibus-mozc (2.28.4715.102) and related dependencies. Fortunately, the dependency chain wasn't that deep. The order I did it was libprotobuf32 (3.21.12-3), libabsl20220623 (20220623.1-1), mozc-server (2.28.4715.102), and finally the ibus-mozc proper.
I haven't tried this version of mozc with the Anki (2.1.15 for jammy) from the official Ubuntu repository though. I guess that it should also work with that too.